The description of the “replacement" of partial thread configuration here can also be formulated as the bulk cell gluing method proposed in [1]}(as we reviewed in Section REF , see also [2]}). Assume that we have found a thread configuration in \(\mathcal {W}\left( T \right)\)  that can lock \(n\)  adjacent elementary regions \(t\)  (denoted as \(t\) =1, 2, ... , \(n\) ) making up \(T\) , as well as a specified set of non-crossing composite regions in \(T\) , and, in particular, the region \(T\)  itself. Note that locking \(T\)  also equivalently means locking its corresponding RT surface \(\tilde{T}\)  or locking its complement \(\bar{T}\) . Then we divide \(\bar{T}\)  into two new elementary regions, \(R\)  and \(S\) . What we need to do is to find a thread configuration for the cell spacetime \(\mathcal {W}\left( {RS} \right)\)  that can further lock \(\left\lbrace  {R,S,\bar{T} = RS} \right\rbrace \) , or its corresponding set of RT surfaces \(\left\lbrace  {\tilde{R},\tilde{S},\tilde{T}} \right\rbrace \)  simultaneously. The bulk cell gluing method points out that, due to the max flow-min cut theorem, on the minimal surface \(\tilde{T}\) , the bit threads must be orthogonal to \(\tilde{T}\)  and the thread density exactly reaches the maximum value \(\frac{1}{{4{G_N}}}\) , therefore, the locking bit thread configurations in two submanifolds \(\mathcal {W}\left( T \right)\)  and \(\mathcal {W}\left( {RS} \right)\)  can just be glued together at the minimal surface \(\tilde{T}\)  due to continuity, and thus providing an overall locking bit thread configuration of the whole spacetime. In [1]}, this gluing process can be achieved unambiguously simply by constructing a special thread configuration described by a \(flow\)  (which is a special case of the \(multiflow\) ) wherein the bit threads are locally parallel everywhere. However, due to the new non-trivial “CFF=PEE” constraint introduced by the holographic qubit threads model, simply using flow to implement gluing will not work in general. The reason is straightforward. In a flow configuration, by definition, the bit threads cannot cross each other. As a result, in the case involving the gluing of a sufficient number of cells, the threads starting from some boundary elementary region may not be able to reach some other distant elementary regions. In other words, this will inevitably make some of the CFFs (component flow fluxes) vanish and possibly violate the “CFF=PEE” condition, which is designed to ensure the bit threads to faithfully represent the entanglement structure of the boundary quantum system. Therefore, we need to propose a more detailed gluing scheme. The key point is to make sure that the bit threads starting from one elementary region can reach any other elementary region at the end.
